Homogeneity and isotropy and derivation of the Lorentz transformations

Question

In deriving the Lorentz transformations I have found (from reading a few different sets lecture notes) that it is argued that they must be linear and thus there general form must be $$x'=Ax+Bt,\quad t'=Dx+Et$$ (assuming relative motion between two inertial frames $S$ and $S'$ along one axis).

My question is, can the linearity of Lorentz transformations be argued purely from Einstein's two postulates, or does one have to assume homogeneity of space and time, and isotropy of space?

I can kind of see that they must be linear purely from the fact that one wishes to map between two inertial frames and hence, in particular, straight lines should be mapped to straight lines (otherwise a particle observed to be unaccelerated in one inertial frame will appear to be accelerating in another). Also, the inverse of a linear transformation is also linear, which is required otherwise such transformations would single it privileged inertial reference frames, violating the principle of relativity.

However, doesn't the mere existence of global inertial frames require spatial homogeneity and isotropy, as otherwise any measurements made by an observer in a given inertial frame would depend on the location of the observer within the inertial frame, and in which direction they make the measurement?!

If one starts off with the assumption of homogeneity and isotropy, then I can definitely see why the transformations should be linear, since homogeneity requires that the form of transformation should not depend on the location of the two inertial frames in space, this the derivative of the transformation should be independent of location, i.e. it should be a constant. Isotropy of space also implies that the transformation should not depend on the relative velocity between the two frames, but at most, the relative speed between them.

I would really appreciate it if someone could enlighten me on this subject?

Answer 1

In this nice reference the author assumes the relativity principle + homogeneity + isotropy and deduces the general coordinate transformations which contain both Lorentz and Galileo transformations. Further he imposes the postulate of the constancy of the speed of light, restricting the transformations to be the Lorentz type.

Answer 2

The answer to your question depends on what you mean by "reference frame" or "local laboratory". Moreover, note that both these two postulates are cast in informal language, so there would be quite a bit of latitude in how they are recast into formal, axiomatic language.

If your local laboratory is big enough to straddle measurable inhomogeneities, then the Galileo relativity principle (of the indetectability of inertial motion from within one's own reference frame and thus the independence of physical law from inertial motion) will detect motion relative those inhomogeneities. Strain gauges in your spaceship that is big enough to feel the tidal effects of a nearby planet as you move past the latter, for example.

Anisotropy is different however - depending on its root cause, it may be independent of motion.

But this is not what people usually mean by the Galileo principle. Before anything else, you must assume a manifold structure for your local bit of spacetime and that transformation between inertial frames can be described by co-ordinate transformations. So the natural framework for a discussion like yours would be in a region of the manifold small enough to behave like the tangent space / small enough to make any inhomogeneities small.

If you assume only manifolds, co-ordinates and co-ordinate transformations wrought by relative motion, then the Galileo principle gives you two things:

1. It shows that the transformation can depend only on the relative velocity;
2. It completes the group structure for the transformations by enforcing associativity.

In fact, to encode the informal Galileo principle into a formal axiom for the purposes of your discussion, you would probably take as an axiom that the transformation should depend only on relative velocity.

You would then have to state your homogeneity assumption as a separate formal axiom but, as we have argued above, you can give an informal justification that both relative velocity alone and homogeneity are contained in the Galileo principle. So the Galileo principle would end up encoded as something like the following axioms:

1. Manifold structure axiom;
2. Transformations between inertial frames are described by co-ordinate transformations;
3. Transformations depend on relative velocity alone;
4. Homogeneity.

You need something else to prove linearity: that the co-ordinate transformations wrought by relative inertial motion are continuous functions of the spacetime co-ordinates. You could include continuity in point 2. above, but I'd argue that this is a little further to the spirit of the Galileo principle - as is clear, it all depends on how you encode the informal statement into axioms.

From the four axioms above, together with continuity of dependence on spacetime co-ordinates, linearity follows, as I discuss in my answer to this question here asked last week.

Once you have linearity (and thus that the transformations are described by a linear matrix group) and the constancy of the speed of light assumption, then still need to assume other things to get to the Lorentz transformation. Constancy of lightspeed essentially means that the eigenvectors of the $2\times2$ matrix of the one-dimensional Lorentz transformation are of the form $(1,\,\pm c)$, which in turn imply a matrix of the form:

$$\Lambda(v)=\left(\begin{array}{cc}\gamma(v)&\delta(v)\\c^2\,\delta(v)&\gamma(v)\end{array}\right)$$

and then you need to assume spatial isotropy, which enforces $\Lambda(-v) = \Lambda(v)^{-1}$ (the so-called relativistic reciprocity principle, and it means that a boosts in an opposite directions are the same transformation with a directional sign change in the appropriate elements). Then you need to assume causality on top of this to get to the Lorentz transformation: see, for example, the "From group postulates" in the "Derivations of the Lorentz transformations" Wikipedia page to see the mechanics of how this can be worked out.

Alternatively, one can forget about the second postulate (constancy of $c$) and instead assume isotropy, continuity of the transformation in its dependence on the relative velocity and causality to infer the existence of a speed which is always measured to be the same in all inertial frames (almost - these assumptions also allow the Galilean transformations, which are excluded by experiment). This is the Ignatowskian approach and I discuss it in my answer here.